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Re: [ontolog-forum] Universal Basic Semantic Structures

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Sun, 30 Sep 2012 13:10:25 -0400
Message-id: <50687D01.9030106@xxxxxxxxxxx>
On 9/30/2012 10:04 AM, Chris Menzel wrote:
> Set theory didn't exist in any systematic form before Cantor and sets
> /per se/ weren't objects of mathematical interest; at most they showed
> up in the analysis of logic, as you note in the cases of Boole and Peirce.    (01)

I dusted off my copy of Boole (1854), and it is rather impressive.
He did not use the word 'set', but he distinguished three terms:    (02)

  1. Propositions -- which can have truth values 1 or 0, but he also
     discussed the use of values between 0 and 1 for probabilities.    (03)

  2. Classes -- which correspond to sets, but without the option
     of considering classes of classes.    (04)

  3. Class terms -- which correspond to the monadic predicates used
     in syllogisms.    (05)

I agree that what is currently called 'set theory' did not exist before
Cantor.  But Boole's treatment is remarkably systematic for such an
early development.    (06)

Many applications of what people call 'sets' don't use anything that
goes beyond Boolean classes or collections.  In fact, John Venn
invented his diagrams to illustrate Boolean operations and their
relationship to syllogisms.    (07)

Most people (and most programmers) who use the word 'set' think
in terms of Venn diagrams, and they never use sets of sets.    (08)

> But pedagogically, it helps to start with Boolean algebra and show
> how many other theories can be developed as specializations of it.    (09)

> I'm not at all sure that's the best way to study set theory; indeed,
> I fear it encourages the confusion between the two conceptions
> of set noted above.    (010)

To avoid confusion, I would reserve the word 'set' for Cantor's version.
But teaching people Boolean algebra is useful for many purposes:    (011)

  1. It shows the underlying commonalities in the operations on
     propositions, predicates, and collections.    (012)

  2. It provides a useful theory of collections that programmers and
     linguists often use, but mistakenly identify with set theory.    (013)

  3. It clarifies the relationships between set theory and mereology.    (014)

As you know, I believe that looking at the lattice of all possible
theories that use a given logic helps clarify seemingly unrelated
issues.  The example of Boolean algebra illustrates that point.    (015)

John    (016)

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